Let $C_0(\mathbb{R})$ denote the analytic functions $f : \mathbb{R} \rightarrow \mathbb{R}$.

I wonder whether there a functions $f \in C_0(\mathbb{R})$ with $f \neq 0$, such that there is a constant $C$, with $$\left| \frac{d^kf(x)}{dx^k} \right | \leq C$$ for all $k \geq 0$, and $\frac{d^kf(x)}{dx^k}$ vanish both at $- \infty$ and $\infty$ for all $k \geq 0$.

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